The third paper in a four-paper series connecting FCC lattice cohomology, Apollonian circle packing, and post-quantum cryptography. Papers 1 and 2 proved that the FCC cycle space carries a natural V4 symmetry and that the FCC primitive cell is the symmetric Apollonian seed in R3,1, with the directed bond map equivariant under both symmetries simultaneously to 8.88x10^-16 across 144 combinations. This paper asks what that object is and why it must be SO(3,1) specifically. The group Gamma generated by the 36 FCC bond transformations is a Kleinian group acting on H3. The Apollonian subgroup A contained in Gamma has limit set conjecturally equal to the Apollonian gasket with Hausdorff dimension 1.3057; the full group Gamma has a strictly larger limit set. Sullivan’s theorem yields a conditional spectral prediction lambda_0 approximately 0.906.This version adds a dimension selection theorem: the Descartes quadratic form for n-dimensional sphere packing has signature (n+1,1) for all n (proved), making Lorentzian geometry a universal packing invariant. However the V4 decomposition of the cycle space exists only in n=3, where it is fully reducible. In n=4 the cycle space is irreducible under S5 and no V4 analogue exists. Three dimensions is not assumed - it is selected by the combined packing-cohomology construction. Three new conjectures are introduced: Self-Replicating Group Structure, Geometric Finiteness, and the Universal Packing Principle. Paper 4 (ADLP post-quantum cryptography) is at zenodo.18826779.